A Supply Chain Network Equilibrium Model with Random Demands

June Dong and Ding Zhang

Department of Marketing and Management

School of Business

State University of New York at Oswego

Oswego, New York 13126

Anna Nagurney∗

Department of Finance and Operations Management

Isenberg School of Management

University of Massachusetts

Amherst, Massachusetts 01003

e-mail: nagurney@gbfin.umass.edu

May 2002; revised October 2002

Appears in European Journal of Operational Research 156 (2004), 194-212.

Abstract:

In this paper, we develop a supply chain network model consisting of manufacturers and

retailers in which the demands associated with the retail outlets are random. We model the

optimizing behavior of the various decision-makers, derive the equilibrium conditions, and

establish the finite-dimensional variational inequality formulation. We provide qualitative

properties of the equilibrium pattern in terms of existence and uniqueness results and also

establish conditions under which the proposed computational procedure is guaranteed to

converge. Finally, we illustrate the model through several numerical examples for which

the equilibrium prices and product shipments are computed. This is the first supply chain

network equilibrium model with random demands for which modeling, qualitative analysis,

and computational results have been obtained.

Key Words: supply chain management, variational inequalities, network equilibrium, ran-

dom demands

∗ corresponding author; phone: 413-545-5635; fax: 413-545-3858; e-mail: nagurney@gbfin.umass.edu

1

1. Introduction

The topic of supply chain modeling and analysis has been of great interest, both from

practical and research perspectives, due to its import in the efficient and cost-effective pro-

duction and flow of goods and services in the network economy. Approaches that have been

utilized to study supply chains have often been multidisciplinary in nature since such mul-

titiered networks of suppliers, manufacturers, retailers, and consumers involve aspects of

manufacturing, transportation and logistics, as well as retailing/marketing.

The body of literature on supply chains is vast (cf. Stadtler and Kilger (2000) and the

references therein) with the associated research being both conceptual in nature (see, e.g.,

Poirier (1996, 1999), Mentzer (2000), Bovet (2000)), due to the complexity of the problem

and the numerous decision-makers in the transactions, as well as analytical (cf. Federgruen

and Zipkin (1986), Federgruen (1993), Slats et al. (1995), Bramel and Simchi-Levi (1997),

Ganeshan et al. (1998), Miller (2001), Hensher, Button, and Brewer (2001) and the references

therein).

Recently, there has been a notable effort expended on the development of decentralized

supply chain network models in which the complexity of the interactions among the various

decision-makers is captured and studied. For example, Lee and Billington (1993) emphasized

the need for the development of decentralized models that allow for a generalized network

structure and simplicity in computation in regards to the study of supply chains. Anupindi

and Bassok (1996), on the other hand, focused on the challenges of systems consisting of

decentralized retailers with information sharing. Lederer and Li (1997), in turn, modeled

the competition among firms that produce goods or services for customers who are sensitive

to delay time.

Nagurney, Dong, and Zhang (2002a) developed a supply chain network equilibrium model

consisting of three tiers of decision-makers on the network and established that the governing

equilibrium conditions which reflected the optimality conditions of the decision-makers con-

sisting of manufacturers, retailers, and consumers along with the market equilibrium condi-

tions could be formulated and studied in a unified manner as a finite-dimensional variational

inequality problem. Such a modeling approach was subsequently extended by Nagurney,

Loo, Dong, and Zhang (2002) to include electronic commerce in the form of business to

2

business (B2B) and business to consumer (B2C) transactions and by Nagurney et al. (2002)

to address the disequilibrium dynamics. More recently, Dong, Zhang, and Nagurney (2002)

introduced multicriteria decision-making into supply chain network equilibrium modeling

and computations. Additional background on related models as well as complementary ones

in finance and in transportation can be found in the book by Nagurney and Dong (2002).

For background on variational inequalities with a special emphasis on network economics,

see the book by Nagurney (1999).

The afore-mentioned variational inequality models of supply chain networks, however,

assumed that the underlying functions, be they, cost, revenue, or profit, were known with

certainty. In this paper, in contrast, we relax this assumption for the demand functions at the

level of the retailers. This result is significant since, in practice, retailers may not know the

demands for a product with certainty but may, nevertheless, possess some information such

as the density function based on historical data and/or forecasted data. Moreover, even with

this extension, we are able to not only derive the optimality conditions for both manufacturers

and the retailers, but also to establish that the governing equilibrium conditions in the

random demand case satisfy a finite-dimensional variational inequality. Moreover, we provide

reasonable conditions on the underlying functions in order to establish qualitative properties

of the equilibrium price and product shipment pattern. Furthermore, we give conditions

that, if satisfied, guarantee convergence of the proposed algorithmic scheme.

We note that Mahajan and Ryzin (2001) considered retailers under uncertain demand and

focused on inventory competition. However, they assumed that the price of the product is

exogenous. In this paper, in contrast, we assume competition, uncertain demand, and provide

a means to determine the equilibrium prices both at the retailers and at the manufacturers.

Lippman and McCardle (1997), in turn, developed a model of inventory competition for

firms but assumed an aggregated random demand. In this paper, we allow each retailer to

handle his own uncertain demand and to engage in competition, which seems closer to actual

practice. More recently, Iida (2002) presented a production-inventory model with uncertain

production capacity and uncertain demand.

The paper is organized as follows. In Section 2, we construct the supply chain network

model with random demands at the retailer tier of nodes. We model the behavior of both the

3

manufacturers and the retailers who are faced with random demands. The manufacturers

are assumed to be profit-maximizers, to produce a homogeneous product, and to compete

with other manufacturers in a noncooperative fashion. They seek to determine their profit-

maximizing outputs and shipments of the product to the retailers. The retailers, who are

faced with random demand for the product at their respective outlets, are also assumed to

be profit-maximizers with a penalty associated with shortage of the product as well as with

excess supply. The retailers also compete with one another in a noncooperative manner.

In Section 2, we derive the optimality conditions and establish the governing equilibrium

concept. We then give the variational inequality formulation, which is utilized in Section 3

to obtain qualitative properties of the equilibrium state as well as properties of the function

that enters the variational inequality required for convergence of the algorithmic scheme.

In Section 4, we outline the algorithm and give convergence results. The algorithm is

then applied in Section 5 to compute the equilibrium price and product shipment pattern

in several supply chain examples. In Section 6 we summarize the results in this paper and

present suggestions for future research.

4

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Figure 1: The Network Structure of the Supply Chain

2. The Supply Chain Network Equilibrium Model with Random Demands

In this Section, we develop the supply chain network equilibrium model with random

demands at the retailer level. The representative decision-makers in our framework are the

manufacturers and the retailers, with the consumers being represented through the random

demands for the product at the different retail outlets. The supply chain network structure

is as depicted in Figure 1.

In particular, we consider m manufacturers involved in the production of a homogeneous

product, which can then be purchased by n retailers, who, in turn, respond to the consumers

via the random demand functions. We denote a typical manufacturer by i and a typical

retailer by j. The supply chain network, as depicted in Figure 1, consists of two tiers of

nodes, with the manufacturers associated with the nodes in the top tier of the supply chain

network and the retailers with the bottom tier of nodes. The links in the supply chain

network denote transportation/transaction links.

We now turn to the discussion of the behavior of the various decision-makers in the supply

chain. We first focus on the manufacturers. We then turn to the retailers.

5

The Manufacturers and their Optimality Conditions

Let qi denote the nonnegative production output of the product by manufacturer i and

group the production outputs of all manufacturers into the column vector q ∈ Rm+ . We

assume that each manufacturer i is faced with a production cost function fi, which can

depend, in general, on the entire vector of production outputs, that is,

fi = fi(q), ∀i. (1)

A manufacturer may ship the product to the retailers, with the amount of the product

shipped (or transacted) between manufacturer i and retailer j denoted by qij. We group

the product shipments between all manufacturers and all retailers into the mn-dimensional

column vector Q.

We associate with each manufacturer and retailer pair (i, j) a transaction cost, denoted

by cij. The transaction cost includes the cost of shipping the product. We assume that the

transaction cost between a manufacturer and retailer pair depends upon the volume of flow

of the product between that pair, that is:

cij = cij(qij), ∀i, j. (2)

The quantity produced by manufacturer i must satisfy the following conservation of flow

equation:

qi =n∑

j=1

qij, (3)

which states that the quantity produced by manufacturer i is equal to the sum of the quan-

tities shipped from the manufacturer to all retailers.

The total costs incurred by a manufacturer i are equal to the sum of his production

cost plus the total transaction costs. His revenue, in turn, is equal to the price that the

manufacturer charges for the product (and the retailers are willing to pay) times the total

quantity purchased of the product from the manufacturer by all the retail outlets. We let

ρ1ij denote the price charged for the product by manufacturer i to retailer j and later in the

paper we discuss how this price, in equilibrium, which is denoted by ρ∗1ij, is arrived at. We

group the prices of the manufacturers into the mn-dimensional column vector ρ1.

6

Noting the conservation of flow equations (3), we can express the criterion of profit

maximization for manufacturer i as:

Maximizen∑

j=1

ρ1ijqij − fi(Q) −n∑

j=1

cij(qij), (4)

subject to qij ≥ 0, for all j.

We assume that the manufacturers compete in a noncooperative fashion. Also, we assume

that the production cost functions and the transaction cost functions for each manufacturer

are continuous and convex. Given that the governing optimization/equilibrium concept

underlying noncooperative behavior is that of Nash (1950, 1951), which states that each

manufacturer will determine his optimal production quantity and shipments, given the opti-

mal ones of the competitors, the optimality conditions for all manufacturers simultaneously

can be expressed as the following variational inequality (cf. Bazaraa, Sherali, and Shetty

(1993), Gabay and Moulin (1980); see also Dafermos and Nagurney (1987) and Nagurney

(1999)): determine Q∗ ∈ Rmn+ satisfying:

m∑

i=1

n∑

j=1

[∂fi(Q

∗)

∂qij

+∂cij(q

∗ij)

∂qij

− ρ1ij

]×[qij − q∗ij

]≥ 0, ∀Q ∈ Rmn

+ . (5)

The optimality conditions as expressed by (5) have a nice economic interpretation, which

is that a manufacturer will ship a positive amount of the product to a retailer (and the flow

on the corresponding link will be positive) if the price that the retailer is willing to pay for

the product is precisely equal to the manufacturer’s marginal production and transaction

cost associated with that retailer. If the sum of the manufacturer’s marginal production and

transaction cost exceeds what the retailer is willing to pay for the product, then there will

be zero shipment of the product between the pair.

The Retailers and their Optimality Conditions

The retailers, in turn, must decide how much to order from the manufacturers in order

to cope with the random demand while still seeking to maximize their profits. A retailer

j is faced with what we term a handling cost, which may include, for example, the display

and storage cost associated with the product. We denote this cost by cj and, in the simplest

7

case, we would have that cj is a function of∑m

i=1 qij, that is, the holding cost of a retailer

is a function of how much of the product he has obtained from the various manufacturers.

However, for the sake of generality, and to enhance the modeling of competition, we allow

the function to, in general, depend also on the amounts of the product held by other retailers

and, therefore, we may write:

cj = cj(Q), ∀j. (6)

Let ρ2j denote the demand price of the product associated with retailer j. We assume

that d̂j(ρ2j) is the demand for the product at the demand price of ρ2j at retail outlet j,

where d̂j(ρ2j) is a random variable with a density function of Fj(x, ρ2j), with ρ2j serving as

a parameter. Hence, we assume that the density function may vary with the demand price.

Let Pj be the probability distribution function of d̂j(ρ2j), that is, Pj(x, ρ2j) = P (d̂j ≤ x) =∫ x0 Fj(x, ρ2j)dx.

Let sj =∑m

i=1 qij, in turn, denote the total supply at retailer j that he obtains from all

the manufacturers. Then, retailer j can sell to the consumers no more than the minimum of

his supply or his demand, that is, the actual sale of j cannot exceed min{sj, d̂j}. Let

∆+j ≡ max{0, sj − d̂j} (7)

and

∆−j ≡ max{0, d̂j − sj}, (8)

where ∆+j is a random variable representing the excess supply (inventory), whereas ∆−

j is a

random variable representing the excess demand (shortage).

Note that the expected values of excess supply and excess demand of retailer j are scalar

functions of sj and ρ2j. In particular, let e+j and e−j denote, respectively, the expected values:

E(∆+j ) and E(∆−

j ), that is,

e+j (sj, ρ2j) ≡ E(∆+

j ) =∫ sj

0(sj − x)Fj(x, ρ2j)dx, (9)

e−j (sj, ρ2j) ≡ E(∆−j ) =

∫ ∞

sj

(x − sj)Fj(x, ρ2j)dx. (10)

8

Assume that the unit penalty of having excess supply at retail outlet j is λ+j and that

the unit penalty of having excess demand is λ−j , where the λ+

j and the λ−j are assumed to

be nonnegative. Then, the expected total penalty of retailer j is given by

E(λ+j ∆+

j + λ−j ∆−

j ) = λ+j e+

j (sj, ρ2j) + λ−j e−j (sj, ρ2j).

Assuming, as already mentioned, that the retailers are also profit-maximizers, the ex-

pected revenue of retailer j is E(ρ2j min{sj, d̂j}). Hence, the optimization problem of a

retailer j can be expressed as:

Maximize E(ρ2j min{sj, d̂j}) − E(λ+j ∆+

j + λ−j ∆−

j ) − cj(Q) −m∑

i=1

ρ1ijqij (11)

subject to:

qij ≥ 0, for all i. (12)

Objective function (11) expresses that the expected profit of retailer j, which is the

difference between the expected revenues and the sum of the expected penalty, the handling

cost, and the payout to the manufacturers, should be maximized.

Applying now the definitions of ∆+j , and ∆−

j , we know that min{sj, d̂j} = d̂j − ∆−j .

Therefore, the objective function (11) can be expressed as

Maximize ρ2jdj(ρ2j)−ρ2je−j (sj, ρ2j)−λ+

j e+j (sj, ρ2j)−λ−

j e−j (sj, ρ2j)−cj(Q)−m∑

i=1

ρ1ijqij (13)

where dj(ρ2j) ≡ E(d̂j) is a scalar function of ρ2j.

We now consider the optimality conditions of the retailers assuming that each retailer

is faced with the optimization problem (11), subject to (12), which represents the nonneg-

ativity assumption on the variables. Here, we also assume that the retailers compete in a

noncooperative manner so that each maximizes his profits, given the actions of the other

retailers. Note that, at this point, we consider that retailers seek to determine the amount

that they wish to obtain from the manufacturers. First, however, we make the following

derivation and introduce the necessary notation:

∂e+j (sj, ρ2j)

∂qij= Pj(sj, ρ2j) = Pj(

m∑

i=1

qij, ρ2j) (14)

9

∂e−j (sj)

∂qij= Pj(sj, ρ2j) − 1 = Pj(

m∑

i=1

qij, ρ2j) − 1. (15)

Assuming that the handling cost for each retailer is continuous and convex, then the

optimality conditions for all the retailers satisfy the variational inequality: determine Q∗ ∈Rmn

+ , satisfying:

m∑

i=1

n∑

j=1

[λ+

j Pj(m∑

i=1

q∗ij, ρ2j) − (λ−j + ρ2j)(1 − Pj(

m∑

i=1

q∗ij, ρ2j)) +∂cj(Q

∗)

∂qij+ ρ1ij

]

×[qij − q∗ij

]≥ 0, ∀Q ∈ Rmn

+ . (16)

In this derivation, as in the derivation of inequality (5), we have not had the prices charged

be variables. They become endogenous variables in the complete supply chain network

equilibrium model.

We now highlight the economic interpretation of the retailers’ optimality conditions. In

inequality (16), we can infer that, if a manufacturer i transacts with a retailer j resulting

in a positive flow of the product between the two, then the selling price at retail outlet j,

ρ2j, with the probability of (1− Pj(∑m

i=1 q∗ij, ρ2j)), that is, when the demand is not less then

the total order quantity, is precisely equal to the retailer j’s payment to the manufacturer,

ρ1ij, plus his marginal cost of handling the product and the penalty of having excess demand

with probability of Pj(∑m

i=1 q∗ij, ρ2j), (which is the probability when actual demand is less

than the order quantity), subtracted by the penalty of having shortage with probability of

(1 − Pj(∑m

i=1 q∗ij, ρ2j)) (when the actual demand is greater than the order quantity).

The Equilibrium Conditions

We now turn to a discussion of the market equilibrium conditions. Subsequently, we

construct the equilibrium conditions for the entire supply chain.

The equilibrium conditions associated with the transactions that take place between the

retailers and the consumers are the stochastic economic equilibrium conditions, which, math-

ematically, take on the following form: For any retailer j; j = 1, . . . , n:

d̂j(ρ∗2j)

{≤ ∑m

i=1 q∗ij a.e., if ρ∗2j = 0

=∑m

i=1 q∗ij a.e., if ρ∗2j > 0,

(17)

10

where a.e. means that the corresponding equality or inequality holds almost everywhere.

Conditions (17) state that, if the equilibrium demand price at outlet j is positive, that

is, ρ∗2j > 0, then the quantities purchased by the retailer from the manufacturers in the

aggregate, that is,∑m

i=1 q∗ij, is equal to the demand, with exceptions of zero probability. These

conditions correspond to the well-known economic equilibrium conditions (cf. Nagurney

(1999) and the references therein). Related equilibrium conditions, but in a deterministic

version, were proposed in Nagurney, Dong, and Zhang (2002a).

Equilibrium conditions (17) are equivalent to the following variational inequality problem,

after taking the expected value and summing over all retailers j: determine ρ∗2 ∈ Rn

+ satisfying

n∑

j=1

(m∑

i=1

q∗ij − dj(ρ∗2j)) ×

[ρ2j − ρ∗

2j

]≥ 0, ∀ρ2 ∈ Rn

+, (18)

where ρ2 is the n-dimensional column vector with components: {ρ21, . . . , ρ2n}.

The Equilibrium Conditions of the Supply Chain

In equilibrium, we must have that the sum of the optimality conditions for all manufac-

turers, as expressed by inequality (5), the optimality conditions for all retailers, as expressed

by inequality (16), and the market equilibrium conditions, as expressed by inequality (18)

must be satisfied. Hence, the shipments that the manufacturers ship to the retailers must

be equal to the shipments that the retailers accept from the manufacturers. We state this

explicitly in the following definition:

Definition 1: Supply Chain Network Equilibrium with Random Demands

The equilibrium state of the supply chain with random demands is one where the product

flows between the two tiers of the decision-makers coincide and the product shipments and

prices satisfy the sum of the optimality conditions (5) and (16) and the conditions (18).

The summation of inequalities (5), (16), and (18) (with the prices at the manufacturers

and the retailers denoted, respectively, by their values at the equilibrium and denoted by ρ∗1

and ρ∗2), after algebraic simplification, yields the following result:

11

Theorem 1: Variational Inequality Formulation

The equilibrium conditions governing the supply chain network model with random demands

are equivalent to the solution of the variational inequality problem given by: determine

(Q∗, ρ∗2)∈Rmn+n

+ satisfying:

m∑

i=1

n∑

j=1

[∂fi(Q

∗)

∂qij

+∂cij(q

∗ij)

∂qij

+∂cj(Q

∗)

∂qij

+ λ+j Pj(

m∑

i=1

q∗ij, ρ∗2j) − (λ−

j + ρ∗2j)(1 − Pj(

m∑

i=1

q∗ij, ρ∗2j))

]

×[qij − q∗ij

]+

n∑

j=1

[m∑

i=1

q∗ij − dj(ρ∗2j)

]×[ρ2j − ρ∗

2j

]≥ 0, ∀(Q, ρ2) ∈ Rmn+n

+ . (19)

For easy reference in the subsequent sections, variational inequality problem (19) can be

rewritten in standard variational inequality form (cf. Nagurney (1999)) as follows:

〈F (X∗)T , X − X∗〉 ≥ 0, ∀X ∈ K ≡ Rmn+n+ , (20)

where X ≡ (Q, ρ2), and F (X) ≡ (Fij, Fj)i=1,...,m;j=1,...,n. The expression 〈·, ·〉 denotes the

inner product in n-dimensional Euclidean space where here N = mn + n.

The variables in the variational inequality problem are: the equilibrium product ship-

ments from the manufacturers to the retailers, Q∗ (from which one can then recover the

production outputs through (3)), and the equilibrium demand prices of the product at the

retailers, ρ∗2.

We now discuss how to recover the prices ρ∗1ij , for all i and j, from the solution of

variational inequality (19). (In Section 4 we describe an algorithm for computing the

solution.) The prices ρ∗1ij (cf. (5)) can be obtained as follows: if q∗ij > 0, then set

ρ∗1ij =

[∂f(Q∗)

∂qij+

∂cij(q∗ij)

∂qij

]; equivalently, (cf. (16)) set ρ∗

1ij = −λ+j Pj(

∑mi=1 q∗ij, ρ

∗2j) + (λ−

j +

ρ∗2j)(1 − Pj(

∑mi=1 q∗ij, ρ

∗2j)) −

∂cj(Q∗)

∂qij.

Note that in this model, the equilibrium prices associated with the manufacturers as well

as those associated with the retailers are endogenous to the model with the manufacturers’

and the retailers’ product shipments at equilibrium being determined at the equilibrium

price vectors.

12

3. Qualitative Properties

In this Section, we provide some qualitative properties of the solution to variational

inequality (19). In particular, we derive existence and uniqueness results. We also investigate

properties of the function F (cf. (20)) that enters the variational inequality of interest here.

Our previous assumptions about the production cost functions, transaction cost functions,

and retailers’ handling cost functions imply that the vector function that enters into the

variational inequality (20) is continuous. However, the feasible set is not compact. Therefore,

we cannot derive the existence of a solution simply from the assumption of continuity of the

functions. Nevertheless, we can impose a rather weak condition to guarantee existence of a

solution pattern.

Let

Kb = {(Q, ρ2)|0 ≤ Q ≤ b1; 0 ≤ ρ2 ≤ b2}, (21)

where b = (b1, b2) ≥ 0 and Q ≤ b1; ρ2j ≤ b2 means that qij ≤ b1 and ρ2j ≤ b2 for all i, j. Then

Kb is a bounded closed convex subset of Rmn+n. Thus, the following variational inequality

〈F (Xb)T , X − Xb〉 ≥ 0, ∀Xb ∈ Kb, (22)

admits at least one solution Xb ∈ Kb, from the standard theory of variational inequalities,

since Kb is compact and F is continuous. Following Kinderlehrer and Stampacchia (1980)

(see also Theorem 1.5 in Nagurney (1999)), we then have:

Theorem 2

Variational inequality (20) admits a solution if there exists a b > 0, such that variational

inequality (22) admits a solution in Kb with

Q1b < b1, ρb2 < b2. (23)

13

Theorem 3: Existence

Suppose that there exist positive constants M, N, R such that: ∀Q with qij ≥ N , ∀i, j:

∂fi(Q)

∂qij+

∂cij(qij)

∂qij+

∂cj(Q)

∂qij+ λ+

j Pj(sj, ρ2j) − (λ−j + ρ2j)(1 − Pj(sj, ρ2j)) ≥ M, (24)

and

dj(ρ2j) ≤ N, ∀ρ2 with ρ2j ≥ R, ∀j. (25)

Then, variational inequality (20) admits at least one solution.

Proof: Follows using analogous arguments as the proof of existence for Proposition 1 in

Nagurney and Zhao (1993) (see also existence proof in Nagurney, Dong, and Zhang (2002b)).

2

Assumptions (24) and (25) can be economically justified as follows. In particular, when

the product shipment, qij, between manufacturer i and retailer j is large, one can expect the

corresponding sum of the marginal costs associated with the production, transaction, and

holding to exceed a positive lower bound, say M . At same time, the large qij causes a greater

sj, which in turn causes the probability distribution Pj(sj, ρ2j) to be close to 1. Consequently,

the sum of the last two terms on the left-hand side of (24), λ+j Pj(sj, ρ2j)−(λ−

j + ρ2j)(1 −P (sj, ρ2j)) is seen to be positive. Therefore, the left-hand side of (24) is greater than or

equal to the lower bound M . On the other hand, a high price ρ2j at retailer j will drive the

demand at that retailer down, in line with the decreasing nature of any demand function,

which ensures (25).

We now recall the concept of an additive production cost, which was introduced by Zhang

and Nagurney (1996) in the stability analysis of dynamic spatial oligopolies, and has also

been employed in the qualitative analysis by Nagurney, Zhang, and Dong (2002b) for the

study of spatial economic networks with multicriteria producers and consumers.

Definition 2: Additive Production Cost

Suppose that for each manufacturer i, the production cost fi is additive, that is

fi(q) = f 1i (qi) + f 2

i (q̄i), (26)

14

where f 1i (qi) is the internal production cost that depends solely on the manufacturer’s own

output level qi, which may include the production operation and the facility maintenance,

etc., and f 2i (q̄i) is the interdependent part of the production cost that is a function of all the

other manufacturers’ output levels q̄i = (q1, · · · , qi−1, qi+1, · · · , qm) and reflects the impact of

the other manufacturers’ production patterns on manufacturer i’s cost. This interdependent

part of the production cost may describe the competition for the resources, consumption of

the homogeneous raw materials, etc.

We now explore additional qualitative properties of the vector function F that enters

the variational inequality problem. Specifically, we show that F is monotone as well as

Lipschitz continuous. These properties are fundamental in establishing the convergence of

the algorithmic scheme in the subsequent section.

Lemma 1

Let g(s, ρ)T = (λ+P (s, ρ) − (λ− + ρ2)(1 − P (s, ρ)), s − d(ρ)), where P is a probability dis-

tribution with the density function of F(x, ρ). Then g(s, ρ) is monotone if and only if

d′(ρ) ≤ −(4αF)−1(P + α∂P∂ρ

)2, where α = λ+ + λ− + ρ.

Proof: In order to prove that g(s, ρ) is monotone with respect to s and ρ, we only need to

show that its Jacobian matrix is positive semidefinite, which will be the case if all eigenvalues

of the symmetric part of the Jacobian matrix are nonnegative real numbers.

Let α = λ+ + λ− + ρ, then the Jacobian matrix of g is

∇g(s, ρ) =

[αF(s, ρ) −1 + P (s, ρ) + α∂P (s,ρ)

∂ρ

1 −d′(ρ)

], (27)

and its symmetric part is

1

2[∇g(s, ρ) + ∇T g(s, ρ)] =

αF(s, ρ), 1

2

(α∂P

∂ρ+ P (s, ρ)

)

12

(α∂P

∂ρ+ P (s, ρ)

), −d′(ρ)

. (28)

The two eigenvalues of (28) are

γmin(s, ρ) =1

2

[(αF − d′) −

√(αF − d′)2 + (α

∂P

∂ρ+ P )2 + 4αFd′

], (29)

15

γmax(s, ρ) =1

2

[(αF − d′) +

√(αF − d′)2 + (α

∂P

∂ρ+ P )2 + 4αFd′

]. (30)

Since what is inside the square root in both (29) and (30) can be rewritten as

(αF + d′)2+

(α

∂P

∂ρ+ P

)2

and can be seen as nonnegative, both eigenvalues are real. Furthermore, under the condition

of the lemma, d′ is non-positive, so the first item in (29) and (30) is nonnegative. The

condition further implies that the second item in (29) and (30), the square root part, is

not greater than the first item, which guarantees that both eigenvalues are nonnegative real

numbers. 2

The condition of Lemma 1 states that the expected demand function of a retailer is a

nonincreasing function with respect to the demand price and its first order derivative has an

upper bound.

Theorem 4: Monotonicity

The function that enters the variational inequality problem (20) is monotone, if the condition

assumed in Lemma 1 is satisfied for each j; j = 1, . . . , n, and if the following conditions are

also satisfied.

Suppose that the production cost functions fi; i = 1, ..., m, are additive, as defined in

Definition 2, and that the f 1i ; i = 1, ..., m, are convex functions. If the cij and cj functions

are convex, for all i, j, then the vector function F that enters the variational inequality (20)

is monotone, that is,

〈(F (X ′) − F (X ′′))T , X ′ − X ′′〉 ≥ 0, ∀X ′, X ′′ ∈ K. (31)

Proof: Let X ′ = (Q′, ρ′2), X ′′ = (Q′′, ρ′′

2). Then, inequality (31) can been seen in the

following deduction:

〈(F (X ′) − F (X ′′))T , X ′ − X ′′〉

16

=m∑

i=1

n∑

j=1

[∂fi(Q

′)

∂qij

− ∂fi(Q′′)

∂qij

]×[q′ij − q′′ij

]

+m∑

i=1

n∑

j=1

[∂cj(Q

′)

∂qij

− ∂cj(Q′′)

∂qij

]×[q′ij − q′′ij

]

+m∑

i=1

n∑

j=1

[∂cij(q

′ij)

∂qij

−∂cij(q

′′ij)

∂qij

]×[q′ij − q′′ij

]

+m∑

i=1

n∑

j=1

[λ+j Pj(

n∑

i=1

q′ij, ρ′2j) − λ+

j Pj(m∑

i=1

q′′ij, ρ′′2j)] × [q′ij − q′′ij]

+m∑

i=1

n∑

j=1

[−λ−j (1 − Pj(

m∑

i=1

q′ij, ρ′2j)) + λ−

j (1 − Pj(m∑

i=1

q′′ij, ρ′′2j))] × [q′ij − q′′ij]

+m∑

i=1

n∑

j=1

[−ρ′2j(1 − Pj(

m∑

i=1

q′ij, ρ′2j)) + ρ′′

2j(1 − Pj(m∑

i=1

q′′ij, ρ′′2j))] × [q′ij − q′′ij]

+n∑

j

[s′j − dj(ρ′2j) − s′′j + dj(ρ

′′2j)] × [ρ′

2j − ρ′′2j]

= (I) + (II) + (III) + (IV ) + (V ) + (V I) + (V II). (32)

Since the fi; i = 1, ..., m, are additive, and the f 1i ; i = 1, ..., m, are convex functions, one

has that

(I) =m∑

i=1

n∑

j=1

[∂f 1

i (Q′)

∂qij− ∂f 1

i (Q′′)

∂qij

]×[q′ij − q′′ij

]≥ 0. (33)

The convexity of cj, for all j, and cij, for all i, j, gives, respectively,

(II) =m∑

i=1

n∑

j=1

[∂cj(Q

′)

∂qij

− ∂cj(Q′)

∂qij

]×[q′ij − q′′ij

]≥ 0 (34)

(III) =m∑

i=1

n∑

j=1

[∂cij(q

′ij)

∂qij

−∂cij(q

′′ij)

∂qij

]×[q′ij − q′′ij

]≥ 0. (35)

Since the probability function Pj is an increasing function, for all j, hence, (IV) and (V) are

greater than or equal to zero.

Let sj =∑m

i=1 qij. Then we have that

(IV ) + (V ) + (V I) + (V II) =

17

+n∑

j=1

[λ+j Pj(s

′j, ρ

′2j) − λ+

j Pj(s′′j , ρ

′′2j)] × [s′j − s′′j ]

+n∑

j=1

[−λ−j (1 − Pj(s

′j, ρ

′2j)) + λ−

j (1 − Pj(s′′j , ρ

′′2j))] × [s′j − s′′j ]

n∑

j=1

[−ρ′2j(1 − Pj(s

′j, ρ

′2j)) + ρ′′

2j(1 − P2j(s′′j , ρ

′′2j))] × [s′j − s′′j ]

+n∑

j=1

[s′j − dj(ρ′2j) − s′′j + dj(ρ

′′2j)] × [ρ′

2j − ρ′′2j]. (36)

Since for each j, applying Lemma 1, we can see that gj(sj, ρ2j) is monotone, hence, (36)

is nonnegative. Therefore, we conclude that (32) is nonnegative in K. The proof is complete.

2

If the conditions in Theorem 4 are slightly strengthened so that the the vector function

enters into the variational inequality problem (20) is strictly monotone, then its solution is

unique (See, e.g., Nagurney (1999)).

Theorem 5: Uniqueness

Suppose that the production cost functions fi; i = 1, ..., m, are additive, as defined in Defini-

tion 2, and that the f 1i ; i = 1, ..., m, are strictly convex functions. If the cij and cj functions

are strictly convex, for all i, j, then the function that enters the variational inequality (20)

has a unique solution in K.

From Theorem 5 it follows that, under the above conditions, the equilibrium product

shipment pattern between the manufacturers and the retailers, as well as the equilibrium

price pattern at the retailers, is unique.

Theorem 6: Lipschitz Continuity

The function that enters the variational inequality problem (20) is Lipschitz continuous, that

is,

‖F (X ′) − F (X ′′)‖ ≤ L‖X ′ − X ′′‖, ∀X ′, X ′′ ∈ K, with L > 0, (37)

under the following conditions:

18

(i). Each fi; i = 1, ..., m, is additive and has a bounded second order derivative;

(ii). The cij and cj have bounded second order derivatives, for all i, j;

Proof: Since the probability function Pj is always less than or equal to 1, for each retailer

j, the result is direct by applying a mid-value theorem from calculus to the vector function

F that enters the variational inequality problem (20). 2

19

4. The Algorithm

In this Section, an algorithm is presented which can be applied to solve any variational

inequality problem in standard form (see (20)), that is:

Determine X∗ ∈ K, satisfying:

〈F (X∗)T , X − X∗〉 ≥ 0, ∀X ∈ K. (38)

The algorithm is guaranteed to converge provided that the function F that enters the vari-

ational inequality is monotone and Lipschitz continuous (and that a solution exists). The

algorithm is the modified projection method of Korpelevich (1977). The statement of the

modified projection method is as follows, where T denotes an iteration counter:

Modified Projection Method

Step 0: Initialization

Set X0 ∈ K. Let T = 1 and let α be a scalar such that 0 < a ≤ 1L, where L is the Lipschitz

continuity constant (cf. Korpelevich (1977)) (see (37)).

Step 1: Computation

Compute X̄T by solving the variational inequality subproblem:

〈(X̄T + aF (XT −1) − XT −1)T , X − X̄T 〉 ≥ 0, ∀X ∈ K. (39)

Step 2: Adaptation

Compute XT by solving the variational inequality subproblem:

〈(XT + aF (X̄T ) − XT −1)T , X − XT 〉 ≥ 0, ∀X ∈ K. (40)

Step 3: Convergence Verification

If max |XTl − XT −1

l | ≤ ε, for all l, with ε > 0, a prespecified tolerance, then stop; else, set

T =: T + 1, and go to Step 1.

20

We now state the convergence result for the modified projection method for this model.

Theorem 7: Convergence

Assume that the function that enters the variational inequality (19) (or (20)) has at least

one solution and satisfies the conditions in Theorem 4 and in Theorem 6. Then the modified

projection method described above converges to the solution of the variational inequality (19)

or (20).

Proof: According to Korpelevich (1977), the modified projection method converges to the

solution of the variational inequality problem of the form (20), provided that the function

F that enters the variational inequality is monotone and Lipschitz continuous and that

a solution exists. Existence of a solution follows from Theorem 3. Monotonicity follows

Theorem 4. Lipschitz continuity, in turn, follows from Theorem 6. 2

We emphasize that, in view of the fact that the feasible set K underlying the supply chain

network equilibrium model with random demands is the nonnegative orthant, the projec-

tion operation encountered in (39) and (40) takes on a very simple form for computational

purposes. Indeed, the product shipments as well as the product prices at a given iteration

in both (39) and in (40) can be exactly and computed in closed form. Hence, the modi-

fied projection method is, in the context of our problem, straightforward to implement. Of

course, one still must determine the step size a, which is fixed, and which depends on the

Lipschitz constant for the particular problem. We return to this point in the subsequent

section in which we present the numerical examples. We note that variants and extensions

of the Korpelevich method for the solution of montonone variational inequalities have been

developed. In particular, we note the method of Khobotov (1987), which provides a rule

for the determination of the step size which is allowed to vary (see also, Marcotte (1991),

Solodov and Tseng (1996), Solodov and Svaiter (1999), and the references therein), but

which also requires the selection of a parameter that is problem dependent.

21

5. Numerical Examples

In this Section, we apply the modified projection method to several numerical examples.

The algorithm was implemented in FORTRAN and the computer system used was a DEC

Alpha system located at the University of Massachusetts at Amherst. The convergence

criterion used was that the absolute value of the product shipments and prices between

two successive iterations differed by no more than 10−4. The parameter a in the modified

projection method (see (39) and (40)) was set to .01 for all the examples. The algorithm

was initialized as follows for all the examples: the initial product shipments were set to zero

whereas the initial demand prices at the retailers were set to one for all the retailers.

In all the examples, we assumed that the demands associated with the retail outlets

followed a uniform distribution. Hence, we assumed that the random demand, d̂j(ρ2j), of

retailer j, is uniformly distributed in [0,bj

ρ2j], bj > 0; j = 1, . . . , n. Therefore,

Pj(x, ρ2j) =xρ2j

bj

, (41)

Fj(x, ρ2j) =ρ2j

bj, (42)

dj(ρ2j) = E(d̂j) =1

2

bj

ρ2j; j = 1, . . . , n. (43)

It is easy to verify that the expected demand function dj(ρ2j) associated with retailer j

is a decreasing function of the price at the demand market.

Example 1

The first numerical supply chain example consisted of two manufacturers and two retailers,

as depicted in Figure 2.

The data for this example were constructed for easy interpretation purposes. The pro-

duction cost functions for the manufacturers were given by:

f1(q) = 2.5q21 + q1q2 + 2q1, f2(q) = 2.5q2

2 + q1q2 + 2q2.

22

Manufacturers

����1 ����

2

����1 ����

2? ?

HHHHHHHHHHHj

������������

Retailers

Figure 2: Supply Chain Network for Numerical Example 1 and its Variants

The transaction cost functions faced by the manufacturers and associated with transacting

with the retailers were given by:

cij(qij) = .5q2ij + 3.5qij, for i = 1, 2; j = 1, 2.

The handling costs of the retailers, in turn, were given by:

c1(Q) = .5(2∑

i=1

qi1)2, c2(Q) = .5(

2∑

i=1

qi2)2.

The bjs were set to 10 for both retail outlets yielding probability distribution functions

as in (41) and the expected demand functions as in (43). The weights (see (13)) associated

with the excess supply and excess demand at the retailers were: λ+j = λ−

j = 1 for j = 1, 2.

Hence, we assigned equal weights for each retailer for excess supply and for excess demand.

The modified projection method converged in 5895 iterations and in a negligible amount

of CPU time and yielded the following equilibrium pattern: the product shipments between

the two manufacturers and the two retailers were: Q∗ : q∗11 = q∗12 = q∗21 = q∗22 = .1590 and

the demand prices at the retailers were: ρ∗21 = ρ∗

22 = 15.2460. It is easy to verify that the

optimality/equilibrium conditions were satisfied with good accuracy.

23

Example 2: Variant 1 of Example 1

We then proceeded to construct a variant of Example 1 as follows. We increased the

bjs associated with both retailers from 10 to 100 but kept the remainder of the data as in

Example 1. In view of (43), this implies that the expected demand associated with each

retailer increased. The structure of the supply chain remained as in Figure 2.

The modified projection method required 4330 iterations for convergence and a negligible

amount of CPU time and yielded the new equilibrium product shipment pattern given by:

Q∗ : q∗11 = q∗12 = q∗21 = q∗22 = .7479 and the new equilibrium demand price pattern given by:

ρ∗21 = ρ∗

22 = 33.2017.

Observe that with a higher bj for each retailer, the product shipments from each manu-

facturer to each retailer increased since the expected demand increased at each outlet and

the demand price at each outlet also increased.

Example 3: Variant 2 of Example 1

To construct Example 3, we kept the data as in Example 1, but now we increased the bjs

even more than they were increased in Example 2. In particular, we now had b1 = b2 = 1000,

which implies (cf. (43)) that the expected demand associated with the retailers was even

higher than in the two preceding examples. Of course, the structure of the supply chain

network remained as in Figure 2.

The modified projection method again converged, in 4345 iterations, and yielded the

equilibrium product shipment pattern given by: Q∗ : q∗11 = q∗12 = q∗21 = q∗22 = 2.7093 and the

equilibrium demand price pattern given by: ρ∗21 = ρ∗

22 = 92.1003.

Note that, in this example, the production outputs of the manufacturers increased since

the demand at the retailers increased as did the demand prices for the product at the retailers.

Example 4

The fourth numerical example (as well as its subsequent variant) consisted of three man-

ufacturers and two retailers. Hence, the supply chain network was now as depicted in Figure

3.

24

Manufacturers

����1 ����

2 ����3

����1 ����

2

AAAAAAU

QQ

QQ

QQ

QQs

��

��

���

AAAAAAU

��

��

���

��

��

��

��+

Retailers

Figure 3: Supply Chain Network for Numerical Example 4 and its Variant

The data for this example were constructed from the data for Example 3, but we added

the necessary functions for the third manufacturer resulting in the following functions:

The production cost functions for the manufacturers were given by:

f1(q) = 2.5q21 + q1q2 + 2q1, f2(q) = 2.5q2

2 + q1q2 + 2q2, f3(q) = .5q23 + .5q1q3 + 2q3.

Note that the production cost function associated with the third manufacturer was distinct

from those of the other two manufacturers.

The transaction cost functions faced by the manufacturers and associated with transacting

with the retailers were given by:

cij(qij) = .5q2ij + 3.5qij, for i = 1, 2, 3; j = 1, 2.

Hence, we retained the transaction cost functions utilized in the preceding three exam-

ples except that we now added new ones associated with the transactions between the new

manufacturer and the two retailers. The handling costs of the retailers remained as in the

preceding examples as did the expected demand functions. This example, hence, illustrates

what may happen when a new manufacturer enters the market with lower production costs

than the other manufacturers.

25

The modified projection method converged in 2122 iterations and an insignificant amount

of CPU time and yielded the following equilibrium pattern: the product shipments be-

tween the three manufacturers and the two retailers were: Q∗ := q∗11 = q∗12 = q∗21 =

q∗22 = 1.3432, q∗31 = q∗32 = 5.3729. The equilibrium demand prices at the two retailers

were: ρ∗21 = ρ∗

22 = 61.9623.

Note that, in comparison to the results in Example 3, with the addition of a new man-

ufacturer, the price charged at the retailer outlets was now lower, due to the competition,

and the increased supply of the product.

Example 5: Variant 1 of Example 4

The fifth numerical example was constructed from the fourth with the data retained but

with the following change: we now increased the weight associated with oversupply at all

retail outlets from 1 to 10. Also, we set the weights associated with undersupply at all retail

outlets to 0. Hence, we now had that λ+j = 10 for j = 1, 2 and λ−

j = 0 for j = 1, 2.

The modified projection method for this example required 2614 iterations for convergence

and yielded the following new equilibrium product shipment pattern: Q∗ := q∗11 = q∗12 = q∗21 =

q∗22 = 1.2303, q∗31 = q∗32 = 4.9211 and the new equilibrium demand prices at the two retailers

were: ρ∗21 = ρ∗

22 = 67.6419.

Hence, when the penalty associated with excess supply increased and there was no penalty

imposed on shortage by each retailer, each retailer reduced his order quantity. The price at

each retailer increased (vis a vis that in Example 4) due to the higher probability of having

a shortage (undersupply of the product).

Example 6

The sixth numerical example consisted of 3 manufacturers and 3 retailers. We retained

the production cost, the transaction cost, and the demand functions as in Example 5 but now

we added data for the third retailer. In particular, we assumed that the transaction costs

associated with transacting with the new retailer were of the same form as given above for

other manufacturer/retailer pairs. In regards to the probability and the demand functions

26

����

����

����

����

����

����

?

@@

@@

@@R

HHHHHHHHHHHj?

@@

@@

@@R

��

��

�� ?

������������

��

��

��

1

1

32

32

Retailers

Manufacturers

Figure 4: The Network Structure of Example 6

for the new retailer, we set b3 = 1000. The weights associated with this retailer were λ+3 = 10

and λ−3 = 0. Hence, these were the same as for the other two retailers in Example 5. The

handling cost function associated with the new retailer was of the same form as for the other

two retailers (and as in the preceding examples).

The supply chain was as depicted in Figure 4.

The modified projection method converged in 3062 iterations and required, as did the

preceding examples, a negligible amount of CPU time, and yielded the equilibrium prod-

uct shipment pattern: Q∗ := q∗11 = 1.3392, q∗12 = .8710, q∗13 = .8710, q∗21 = 1.3392, q∗22 =

.8710, q∗23 = .8710, q∗31 = 4.0668, q∗32 = 5.0759, q∗33 = 5.0759, and the equilibrium demand

prices at the three retailers: ρ∗21 = 74.0142, ρ∗

22 = ρ∗23 = 73.2373.

It is interesting to note that the total quantity that each retailer now ordered decreased

from the amount ordered in Example 5. For example, here retailer 1 ordered∑3

i=1 q∗i1 =

7.3811, whereas in Example 5 he had ordered (in equilibrium) the amount 6.8. This effect is

reasonable since when there are more resources available (in the form of more manufacturers

more producing a product) less has to be stored.

These examples illustrate the variety of scenarios that can be evaluated in regards to

supply chain network problems with random demands. Indeed, one can vary the weights

27

associated with the retailers, the number of manufacturers and/or retailers, as well as the

parameters in the distribution (and demand) functions and, through the application of the

computational procedure, evaluate the effects on the equilibrium product shipments and

prices.

We emphasize the following features of the modified projection method of Korpelevich in

the context or our supply chain model:

1. the method is easy to implement and results in closed form expressions that are efficiently

solved using explicit formulae;

2. the method is robust and with the convergence tolerance used provided equilibrium

solutions to good accuracy, and

3. the same step size, albeit small, was used for all the examples, and was selected without

any precomputations; the step size was selected to be small to guarantee that it would satisfy

the condition 0 < a ≤ 1L, as needed to guarantee convergence.

Of course, the method did require many iterations but the overall CPU time was negligible

since the subproblems were so simple computationally. It would be interesting to investigate

the application of extensions of the modified projection method with varying step sizes

(including that of Khobotov’s) to this (as well as other) supply chain network equilibrium

models. Nevertheless, we agree with Solodov and Tseng (1996) in that the Korpelevich

method “is a very practical method” for solving a variational inequality problem and, in

particular, our supply chain model with random demands.

28

6. Summary and Conclusions

In this paper, we have proposed a theoretically rigorous framework for the modeling,

qualitative analysis, and computation of solutions to supply chain network problems within

an equilibrium context in the case of random demands associated with the retailers. The

theoretical analysis is based on finite-dimensional variational inequality theory. Before the

results obtained in the paper, the modeling and analysis of supply chain network problems

using variational inequality theory assumed that all the underlying functions were known

were certainty.

In particular, we assumed a supply chain consisting of competing manufacturers and

competing retailers, each of whom seeks to maximize profits. The retailers are faced with

random demands at their outlets and also penalize excess supply (inventory) and excess

demand (shortage) at their particular outlets individually. We derived the governing equi-

librium conditions and then showed that they satisfy a variational inequality problem. The

variational inequality was then utilized to obtain, under reasonable conditions, existence of

the equilibrium product and price pattern, as well as uniqueness. Moreover, we established

additional properties of the function that enters the variational inequality that were then

utilized to establish convergence of the proposed algorithmic scheme.

We then applied the computational procedure, which in the context of our model has the

attractive feature that it yields subproblems in prices and product shipments that can be

solved in closed form, to several numerical supply chain examples. The numerical examples

illustrate the flexibility of the model.

This work establishes the foundations for decentralized and competitive supply chain

network problems in the case of random demands within an equilibrium framework. Future

research may include the modeling of random costs as well as the modeling of disequiibrium

dynamics.

29

Acknowledgments

The research of the first and third authors was supported, in part, by NSF Grant No.:

IIS-0002647. The research of the third author was also supported, in part, by NSF Grant No.:

CMS-0085720 and by NSF Grant No.: INT-0000309. The support is gratefully acknowledged.

The authors are grateful to the two anonymous reviewers for helpful comments and sug-

gestions.

30

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